cp's OEIS Frontend

A336014 Irregular triangle read by rows: T(n,1) = T(n,2) = T(n,3*n-2) = T(n,3*n-1) = n for n >= 1 and T(n,k) = T(n-1,k-2) + T(n-1,k-1) for n > 1, 3 <= k <= 3*(n-1).

%I A336014 #8 Sep 14 2020 12:18:00
%S A336014 1,1,2,2,2,2,2,3,3,4,4,4,4,3,3,4,4,6,7,8,8,8,7,6,4,4,5,5,8,10,13,15,
%T A336014 16,16,15,13,10,8,5,5,6,6,10,13,18,23,28,31,32,31,28,23,18,13,10,6,6,
%U A336014 7,7,12,16,23,31,41,51,59,63,63,59,51,41,31,23,16,12,7,7
%N A336014 Irregular triangle read by rows: T(n,1) = T(n,2) = T(n,3*n-2) = T(n,3*n-1) = n for n >= 1 and T(n,k) = T(n-1,k-2) + T(n-1,k-1) for n > 1, 3 <= k <= 3*(n-1).
%C A336014 The number of terms in row n is 3*n-1 = A016789(n-1).
%C A336014 The sum of row n is equal to 2*A094002(n-1) = 2*A188589(n).
%C A336014 Fibonacci(n) = T(n+k,n) - T(n+k-1,n) for n >= 1, k = 1,2,3,...
%C A336014 The elements b(k) of the main diagonal, superdiagonal 1 and all subdiagonals have the recursive formula: b(k) = 2*b(k-1) + b(k-2) - 2*b(k-3) - b(k-4) for k > 4.
%F A336014 T(n,k) = T(n,3*k-n) for 1 <= k <= 3*n-1.
%F A336014 T(n,k) = Sum_{u=2*(n-k)+3..2*n-k+1} ceiling(u/2)*A065941(k-2,u-2*(n-k)-3) for n >= 3, 3 <= k <= n.
%F A336014 T(n,k) = Sum_{m1=1..k-n} A208354(m1)*binomial(n-m1-1, k-n-m1) + Sum_{m2=1..2*n-k} A208354(m2)*binomial(n-m2-1, 2*n-k-m2) for n >= 2, n+1 <= k <= 2*n-1.
%F A336014 T(n,k) = Sum_{u=2*(k-2*n)+3..k-n+1} ceiling(u/2)*A065941(3*n-k-2,u-2*(k-2*n)-3) for n>= 3, 2*n <= k <= 3*(n-1).
%F A336014 T(n,k) = A208354(k) + (n-k)*Fibonacci(k) for n >= 3, 3 <= k <= n.
%F A336014 T(n,k) = A029907(k-1) + (n-k+1)*Fibonacci(k) for n >= 2, 3 <= k <= n+1.
%e A336014 Triangle begins:
%e A336014 n\k 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20...
%e A336014 1   1  1
%e A336014 2   2  2  2  2  2
%e A336014 3   3  3  4  4  4  4  3  3
%e A336014 4   4  4  6  7  8  8  8  7  6  4  4
%e A336014 5   5  5  8 10 13 15 16 16 15 13 10  8  5  5
%e A336014 6   6  6 10 13 18 23 28 31 32 31 28 23 18 13 10  6  6
%e A336014 7   7  7 12 16 23 31 41 51 59 63 63 59 51 41 31 23 16 12  7  7
%e A336014 ...
%Y A336014 Superdiagonal 1 is A029907 for n >= 1.
%Y A336014 The main diagonal is A208354 for n >= 1.
%Y A336014 Subdiagonal 1 is A102702(n-1) for n >= 1.
%Y A336014 Subdiagonal 2 is A206268(n+2) for n >= 1 (conjectured).
%Y A336014 Subdiagonal 3 is A191830(n+3) for n >= 1.
%Y A336014 Cf. A007318, A051597, A228196.
%K A336014 nonn,tabf
%O A336014 1,3
%A A336014 _Lechoslaw Ratajczak_, Jul 04 2020